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Budget Feasible Mechanisms for Experimental Design Thibaut Horel Joint work with Stratis Ioannidis and S. Muthukrishnan February 26, 2013 Thibaut Horel () February 26, 2013 1 / 1 Motivation Data mining engine Thibaut Horel () February 26,

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Budget Feasible Mechanisms for Experimental Design

Thibaut Horel Joint work with Stratis Ioannidis and S. Muthukrishnan February 26, 2013

Thibaut Horel () February 26, 2013 1 / 1

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SLIDE 2

Motivation

Data mining engine

Thibaut Horel () February 26, 2013 2 / 1

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SLIDE 3

Motivation

Data mining engine

c1$ c 3 $ c2$

Thibaut Horel () February 26, 2013 2 / 1

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SLIDE 4

Motivation

Data mining engine

c1$ c 3 $ c2$ p1$ p 2 $

B$

Thibaut Horel () February 26, 2013 2 / 1

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SLIDE 5

Motivation

Data mining engine

p1$ p 2 $

B$

Thibaut Horel () February 26, 2013 2 / 1

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SLIDE 6

Challenges

Value of data? How to optimize it? Strategic users?

Thibaut Horel () February 26, 2013 3 / 1

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SLIDE 7

Challenges

Value of data? How to optimize it? Strategic users?

Thibaut Horel () February 26, 2013 3 / 1

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SLIDE 8

Challenges

Value of data? How to optimize it? Strategic users?

Thibaut Horel () February 26, 2013 3 / 1

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SLIDE 9

Challenges

Value of data? How to optimize it? Strategic users?

Thibaut Horel () February 26, 2013 3 / 1

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SLIDE 10

Contributions

case of the linear regression deterministic mechanism generalization (randomized mechanism)

Thibaut Horel () February 26, 2013 4 / 1

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SLIDE 11

Contributions

case of the linear regression deterministic mechanism generalization (randomized mechanism)

Thibaut Horel () February 26, 2013 4 / 1

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SLIDE 12

Contributions

case of the linear regression deterministic mechanism generalization (randomized mechanism)

Thibaut Horel () February 26, 2013 4 / 1

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SLIDE 13

Contributions

case of the linear regression deterministic mechanism generalization (randomized mechanism)

Thibaut Horel () February 26, 2013 4 / 1

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SLIDE 14

Outline

Thibaut Horel () February 26, 2013 5 / 1

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Outline

Thibaut Horel () February 26, 2013 6 / 1

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Reverse auction

set of N sellers: A = {1, . . . , N}; a buyer V value function of the buyer, V : 2A → R+ ci ∈ R+ price of seller’s i good B budget constraint of the buyer

Goal

Find S ⊂ A maximizing V (S) Find payment pi to seller i ∈ S

Thibaut Horel () February 26, 2013 7 / 1

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SLIDE 17

Reverse auction

set of N sellers: A = {1, . . . , N}; a buyer V value function of the buyer, V : 2A → R+ ci ∈ R+ price of seller’s i good B budget constraint of the buyer

Goal

Find S ⊂ A maximizing V (S) Find payment pi to seller i ∈ S

Thibaut Horel () February 26, 2013 7 / 1

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SLIDE 18

Reverse auction

set of N sellers: A = {1, . . . , N}; a buyer V value function of the buyer, V : 2A → R+ ci ∈ R+ price of seller’s i good B budget constraint of the buyer

Goal

Find S ⊂ A maximizing V (S) Find payment pi to seller i ∈ S

Thibaut Horel () February 26, 2013 7 / 1

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SLIDE 19

Reverse auction

set of N sellers: A = {1, . . . , N}; a buyer V value function of the buyer, V : 2A → R+ ci ∈ R+ price of seller’s i good B budget constraint of the buyer

Goal

Find S ⊂ A maximizing V (S) Find payment pi to seller i ∈ S

Thibaut Horel () February 26, 2013 7 / 1

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SLIDE 20

Reverse auction

set of N sellers: A = {1, . . . , N}; a buyer V value function of the buyer, V : 2A → R+ ci ∈ R+ price of seller’s i good B budget constraint of the buyer

Goal

Find S ⊂ A maximizing V (S) Find payment pi to seller i ∈ S

Thibaut Horel () February 26, 2013 7 / 1

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SLIDE 21

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 22

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 23

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 24

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 25

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 26

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 27

Objectives

Payments (pi)i∈S must be: individually rational: pi ≥ ci, i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible:

i∈S pi ≤ B

Mechanism must be: computationally efficient: polynomial time good approximation: V (OPT) ≤ αV (S) with: OPT = arg max

S⊂A

  • V (S) |
  • i∈S

ci ≤ B

  • Thibaut Horel ()

February 26, 2013 8 / 1

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SLIDE 28

Known results

When V is submodular: randomized budget feasible mechanism, approximation ratio: 7.91 (Chen et al., 2011) deterministic mechanisms for:

◮ Knapsack: 2 +

√ 2 (Chen et al., 2011)

◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

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Known results

When V is submodular: randomized budget feasible mechanism, approximation ratio: 7.91 (Chen et al., 2011) deterministic mechanisms for:

◮ Knapsack: 2 +

√ 2 (Chen et al., 2011)

◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

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Known results

When V is submodular: randomized budget feasible mechanism, approximation ratio: 7.91 (Chen et al., 2011) deterministic mechanisms for:

◮ Knapsack: 2 +

√ 2 (Chen et al., 2011)

◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

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Outline

Thibaut Horel () February 26, 2013 10 / 1

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Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

N users

Thibaut Horel () February 26, 2013 11 / 1

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Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

xi: public features (e.g. age, gender, height, etc.)

Thibaut Horel () February 26, 2013 11 / 1

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Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

yi: private data (e.g. disease, etc.)

Thibaut Horel () February 26, 2013 11 / 1

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SLIDE 35

Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

Gaussian Linear model: yi = βTxi + εi β∗ = arg min

β

  • i

|yi − βTxi|2

Thibaut Horel () February 26, 2013 11 / 1

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SLIDE 36

Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

c1$ c 3 $ c2$

Thibaut Horel () February 26, 2013 11 / 1

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Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

c1$ c 3 $ c2$ p1$ p 2 $

B$

Thibaut Horel () February 26, 2013 11 / 1

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SLIDE 38

Linear Regression

Linear regression

x1 x2 x3 y2 y1 y3

c1$ c 3 $ c2$ p1$ p 2 $

B$

Thibaut Horel () February 26, 2013 11 / 1

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Experimental design

Public vector of features xi ∈ Rd Private data yi ∈ R Gaussian linear model: yi = βTxi + εi, β ∈ Rd, εi ∼ N(0, σ2) Which users to select? Experimental design ⇒ D-optimal criterion

Experimental Design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to

|S| ≤ k

Thibaut Horel () February 26, 2013 12 / 1

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Experimental design

Public vector of features xi ∈ Rd Private data yi ∈ R Gaussian linear model: yi = βTxi + εi, β ∈ Rd, εi ∼ N(0, σ2) Which users to select? Experimental design ⇒ D-optimal criterion

Experimental Design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to

|S| ≤ k

Thibaut Horel () February 26, 2013 12 / 1

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SLIDE 41

Experimental design

Public vector of features xi ∈ Rd Private data yi ∈ R Gaussian linear model: yi = βTxi + εi, β ∈ Rd, εi ∼ N(0, σ2) Which users to select? Experimental design ⇒ D-optimal criterion

Experimental Design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to

|S| ≤ k

Thibaut Horel () February 26, 2013 12 / 1

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SLIDE 42

Budgeted Experimental design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to
  • i∈S

ci ≤ B the non-strategic optimization problem is NP-hard V is submodular previous results give a randomized budget feasible mechanism deterministic mechanism?

Thibaut Horel () February 26, 2013 13 / 1

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Budgeted Experimental design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to
  • i∈S

ci ≤ B the non-strategic optimization problem is NP-hard V is submodular previous results give a randomized budget feasible mechanism deterministic mechanism?

Thibaut Horel () February 26, 2013 13 / 1

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SLIDE 44

Budgeted Experimental design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to
  • i∈S

ci ≤ B the non-strategic optimization problem is NP-hard V is submodular previous results give a randomized budget feasible mechanism deterministic mechanism?

Thibaut Horel () February 26, 2013 13 / 1

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SLIDE 45

Budgeted Experimental design

maximize V (S) = log det

  • Id +
  • i∈S

xixT

i

  • subject to
  • i∈S

ci ≤ B the non-strategic optimization problem is NP-hard V is submodular previous results give a randomized budget feasible mechanism deterministic mechanism?

Thibaut Horel () February 26, 2013 13 / 1

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SLIDE 46

Main result

Theorem

There exists a budget feasible, individually rational and truthful mechanism for budgeted experimental design which runs in polynomial time. Its approximation ratio is: 10e − 3 + √ 64e2 − 24e + 9 2(e − 1) ≃ 12.98

Thibaut Horel () February 26, 2013 14 / 1

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SLIDE 47

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ V (OPT−i∗) ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 48

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ V (OPT−i∗) ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 49

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ V (OPT−i∗) ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 50

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ L∗ ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 51

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ L∗ ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 52

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ L∗ ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 53

Sketch of proof

Mechanism (Chen et. al, 2011) for submodular V

Find i∗ = arg maxi V ({i}) Compute SG greedily Return:

◮ {i∗} if V ({i∗}) ≥ L∗ ◮ SG otherwise

Valid mechanism, approximation ratio: 8.34 Problem: OPT−i∗ is NP-hard to compute Solution: Replace V (OPT−i∗) with L∗: computable in polynomial time close to V (OPT−i∗) Knapsack (Chen et al., 2011) Coverage (Singer, 2012)

Thibaut Horel () February 26, 2013 15 / 1

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SLIDE 54

Sketch of proof (2)

L∗ = arg max

λ∈[0,1]n

  • log det
  • Id +
  • i

λixixT

i

  • |

n

  • i=1

λici ≤ B

  • polynomial time? convex optimization problem

close to V (OPT−i∗)?

Technical lemma

L∗ ≤ 2V (OPT) + V ({i∗})

Thibaut Horel () February 26, 2013 16 / 1

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SLIDE 55

Sketch of proof (2)

L∗ = arg max

λ∈[0,1]n

  • log det
  • Id +
  • i

λixixT

i

  • |

n

  • i=1

λici ≤ B

  • polynomial time? convex optimization problem

close to V (OPT−i∗)?

Technical lemma

L∗ ≤ 2V (OPT) + V ({i∗})

Thibaut Horel () February 26, 2013 16 / 1

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SLIDE 56

Sketch of proof (2)

L∗ = arg max

λ∈[0,1]n

  • log det
  • Id +
  • i

λixixT

i

  • |

n

  • i=1

λici ≤ B

  • polynomial time? convex optimization problem

close to V (OPT−i∗)?

Technical lemma

L∗ ≤ 2V (OPT) + V ({i∗})

Thibaut Horel () February 26, 2013 16 / 1

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SLIDE 57

Sketch of proof (2)

L∗ = arg max

λ∈[0,1]n

  • log det
  • Id +
  • i

λixixT

i

  • |

n

  • i=1

λici ≤ B

  • polynomial time? convex optimization problem

close to V (OPT−i∗)?

Technical lemma

L∗ ≤ 2V (OPT) + V ({i∗})

Thibaut Horel () February 26, 2013 16 / 1

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SLIDE 58

Outline

Thibaut Horel () February 26, 2013 17 / 1

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SLIDE 59

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 60

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 61

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 62

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 63

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 64

Generalization

Generative model: yi = f (xi) + εi, i ∈ A prior knowledge of the experimenter: f is a random variable uncertainty of the experimenter: entropy H(f ) after observing {yi, i ∈ S}, uncertainty: H(f | S)

Value function: Information gain

V (S) = H(f ) − H(f | S), S ⊂ A V is submodular ⇒ randomized budget feasible mechanism

Thibaut Horel () February 26, 2013 18 / 1

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SLIDE 65

Conclusion

Experimental design + Auction theory = powerful framework deterministic mechanism for the general case? other learning tasks? approximation ratio ≃ 13. Lower bound: 2

Thibaut Horel () February 26, 2013 19 / 1

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SLIDE 66

Conclusion

Experimental design + Auction theory = powerful framework deterministic mechanism for the general case? other learning tasks? approximation ratio ≃ 13. Lower bound: 2

Thibaut Horel () February 26, 2013 19 / 1

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SLIDE 67

Conclusion

Experimental design + Auction theory = powerful framework deterministic mechanism for the general case? other learning tasks? approximation ratio ≃ 13. Lower bound: 2

Thibaut Horel () February 26, 2013 19 / 1